Subgroups — C₂²×C₇×#11765917 #44544690

Order 32004.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#11765917	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₇×C₁₂₇	C₃	1	1	yes	maximal, characteristic
3	C₇×#11765917	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₃×C₇×C₁₂₇	C₂×C₃	1	3	yes	—
5	C₂²×#11765917	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₇×C₁₂₇	C₉	1	1	yes	characteristic, complemented, Hall
7	C₃×C₇×C₁₂₇	C₂²×C₃	1	1	yes	characteristic
8	C₂×#11765917	C₂×C₇	1	3	yes	direct product factor
9	C₂×C₇×C₁₂₇	C₂×C₉	1	3	yes	complemented
10	C₂²×C₃×C₁₂₇	C₃×C₇	1	1	yes	characteristic
11	#11765917	C₂²×C₇	1	1	yes	characteristic, direct product factor, Hall
12	C₇×C₁₂₇	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃×C₁₂₇	C₂×C₃×C₇	1	3	yes	—
14	C₂²×C₁₂₇	C₇×C₉	1	1	yes	characteristic, complemented, Hall
15	C₃×C₁₂₇	C₂²×C₃×C₇	1	1	yes	characteristic
16	C₂×C₁₂₇	C₂×C₇×C₉	1	3	yes	complemented
17	C₂²×C₇×C₉	—	127	1	no	maximal, Hall
18	C₁₂₇	C₂²×C₇×C₉	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₇×C₉	—	127	3	no	—
20	C₂²×C₃×C₇	C₁₂₇⋊C₃	1	1	yes	characteristic
21	C₇×C₉	—	127	1	no	Hall
22	C₂×C₃×C₇	C₂×C₁₂₇⋊C₃	1	3	yes	—
23	C₂²×C₉	—	127	1	no	Hall
24	C₂²×C₇	#11765917	1	1	yes	characteristic, direct product factor, Hall
25	C₃×C₇	C₂²×C₁₂₇⋊C₃	1	1	yes	characteristic
26	C₂×C₉	—	127	3	no	—
27	C₂×C₇	C₂×#11765917	1	3	yes	direct product factor
28	C₂²×C₃	C₇×C₁₂₇⋊C₃	1	1	yes	characteristic
29	C₉	—	127	1	no	Sylow
30	C₇	C₂²×#11765917	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃	C₂×C₇×C₁₂₇⋊C₃	1	3	yes	—
32	C₂²	C₇×#11765917	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	C₂²×C₇×C₁₂₇⋊C₃	1	1	yes	characteristic
34	C₂	C₂×C₇×#11765917	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₇	C₁₂₇⋊C₃
commutator, hypocenter	C₁₂₇	C₂²×C₇×C₉
Frattini	C₃	C₂²×C₇×C₁₂₇⋊C₃
Fitting, socle, prime radical	C₂²×C₃×C₇×C₁₂₇	C₃
cosocle	C₃×C₁₂₇	C₂²×C₃×C₇
LCS	G ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₃×C₇

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₇×#11765917	C₂	direct (N × H)
3. C₇×#11765917	C₂²	direct (N × H)
5. C₂²×#11765917	C₇	direct (N × H)
6. C₂²×C₇×C₁₂₇	C₉	semidirect
8. C₂×#11765917	C₂×C₇	direct (N × H)
9. C₂×C₇×C₁₂₇	C₂×C₉	semidirect
11. #11765917	C₂²×C₇	direct (N × H)
12. C₇×C₁₂₇	C₂²×C₉	semidirect
14. C₂²×C₁₂₇	C₇×C₉	semidirect
16. C₂×C₁₂₇	C₂×C₇×C₉	semidirect
18. C₁₂₇	C₂²×C₇×C₉	semidirect
24. C₂²×C₇	#11765917	direct (N × H)
27. C₂×C₇	C₂×#11765917	direct (N × H)
30. C₇	C₂²×#11765917	direct (N × H)
32. C₂²	C₇×#11765917	direct (N × H)
34. C₂	C₂×C₇×#11765917	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#11765917	2
C₂²×C₃×C₇×C₁₂₇	3
C₂²×#11765917	7
C₂²×C₇×C₉	127

Subgroups — C22×C7×#11765917 #44544690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#11765917 #44544690